There are versions of the CLR inequality and the LT inequalities for fractional Schrödinger operators \(\Delta ^{n/2p}+V\)  (see, e.g., [1]}, [2]}, [3]}, [4]}, [5]}, [6]}, [7]}). Such operators naturally appear in the framework of fractional quantum mechanics [8]}; for \(p=n\)  we get the ultra-relativistic Schrödinger operator \(|\nabla |+V\)  (see [1]}). We refer to the recent survey [10]} and the book [11]} (and the numerous references therein) for a more thorough account on the CLR and LT inequalities and their various applications and generalizations.
